Building an end-to-end conversational agent for multi-domain task-oriented dialogues has been an open challenge for two main reasons. First, tracking dialogue states of multiple domains is non-trivial as the dialogue agent must obtain complete states from all relevant domains, some of which might have shared slots among domains as well as unique slots specifically for one domain only. Second, the dialogue agent must also process various types of information across domains, including dialogue context, dialogue states, and database, to generate natural responses to users. Unlike the existing approaches that are often designed to train each module separately, we propose UniConv -- a novel unified neural architecture for end-to-end conversational systems in multi-domain task-oriented dialogues, which is designed to jointly train (i) a Bi-level State Tracker which tracks dialogue states by learning signals at both slot and domain level independently, and (ii) a Joint Dialogue Act and Response Generator which incorporates information from various input components and models dialogue acts and target responses simultaneously. We conduct comprehensive experiments in dialogue state tracking, context-to-text, and end-to-end settings on the MultiWOZ2.1 benchmark, achieving superior performance over competitive baselines.